MaBloWriMo 9: omega and its ilk

So far, we have defined a sequence of numbers s_n, and showed that

s_n = \omega^{2^n} + \overline{\omega}^{2^n}

where \omega = 2 + \sqrt 3 and \overline{\omega} = 2 - \sqrt 3. This is a big step: the s_n are defined recursively (that is, each s_n is defined in terms of the previous s_{n-1}), but \omega and \overline{\omega} give us a direct arithmetic expression for s_n, which can make proving things about the s_n easier. So our next goal will be to better understand \omega.

As a warmup, consider that \omega and \overline{\omega} are both of the form a + b\sqrt 3, where a and b are integers. Consider the set of all such numbers, call it S. So, for example, S contains things like 4 - 19\sqrt{3} and 0 + \sqrt{3} and 99 + 999\sqrt{3} and 2 + 0\sqrt{3}… and so on. (Mathematicians would call this set \mathbb{Z}[\sqrt{3}], but it doesn’t really matter what we call it.) First of all, note that if we take any two numbers in S and add them, we get another number in S:

(a + b\sqrt{3}) + (c + d\sqrt{3}) = (a + c) + (b + d)\sqrt{3}

For example, (2 + 3\sqrt{3}) + (4 - 4\sqrt{3}) = 6 - \sqrt{3}. A shorter way to say this is that S is “closed under addition”. The additive identity, 0, is also a member of S, since 0 = 0 + 0\sqrt{3}.

We can show that S is closed under multiplication, too:

(a + b\sqrt{3})(c + d\sqrt{3}) = ac + bc\sqrt{3} + ad\sqrt{3} + 3bd = (ac + 3bd) + (bc + ad)\sqrt{3}

And the multiplicative identity 1 is likewise in S, since 1 = 1 + 0 \sqrt{3}.

So S is a sort of “system of numbers” in a similar sense to the integers, or rational numbers, or especially the complex numbers a + bi. At this point we could start talking about rings and fields and stuff, but we don’t actually need anything that complicated. We just need to talk a little bit about groups. I’ve been wanting to write about groups on this blog for a while, and this is a great excuse! So, tomorrow, we’ll start to learn about groups and develop some of the ideas we will need to understand \omega.

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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